Physica B 191 (1993) 210-216 North-Holland

SDI: 0921-4526(93)E0139-8

PHYSICA

Thermo- and magnetic-field-induced dynamics of ferroelectric interphase boundaries

A. Gordon Department of Mathematics and Physics, Haifa University at Oranim, Israel

I .D. Vagner and P. Wyder Max-Planck-lnstitut far Festk6rperforschung, Hochfeld Magnetlabor, Grenoble Cedex 9, France

Received 10 May 1993

Thermo- and magnetic-field-induced dynamics of first-order ferroelectric and antiferroelectric phase transitions are considered, taking into account the inertia effect. The width and velocity of interphase boundaries are calculated as functions of temperature and magnetic field strength. The results obtained here are essentially different from those of the Ginzburg-Landau theory and of the case of the small kinetic energy term, and they may also be used for the description of the kinetics of ferroelectric phase transitions in high-temperature superconductive perovskites.

I. Introduction

In recent years the investigation of the domain and interphase boundary dynamics in ferroelec- trics and antiferroelectrics has received renewed experimental and theoretical interest [1-6]. This was motivated by the discovery of ferroelectric properties in the high-temperature superconduct- ing perovskites [7-12] and by the extensive research in the metastable states in condensed matter physics [13-26]. As has recently been shown [27], magnetic fields can move ferroelec- tric and antiferroelectric interphase boundaries leading to the development of the new phase at the expense of the parent one. The propagation of such interphase boundaries in electrically ordered substances is expected to be governed by the magnetoelectric effect [28-30]. The straightforward magnetic field influence on elec- tric polarization has recently been measured in

Correspondence to: A. Gordon, Department of Mathematics and Physics, Haifa University at Oranim, 36910 Tivon, Israel.

ferroelectric paramagnetic rare-earth molybdates and ferroelectric liquid crystals [31,32]. First measurements of the magnetic field effects on the phase transition temperature in ferroelectric perovskites was made in refs. [33,34]. The mag- netic field effect in ferroelectrics is probably caused by the coupling of the magnetic field with the orbital wave functions determining the change in polarizability. Until now the study of the temperature and magnetic field dynamics of ferroelectric interphase boundaries has been based on the Ginzburg-Landau approach giving only the overdamped movement of the inter- phase boundaries. In this paper we revise the theory of thermo- and magnetic-field-induced dynamics of ferroelectric boundaries taking into account their faster, underdamped motion and thereby improve the known Ginzburg-Landau approach [2] for the description of the interphase boundaries dynamics. To describe relatively fast motions of the interphase boundaries, we take into consideration the inertia effect decreasing the role of the damping term in the Ginzburg- Landau dynamic theory. The behaviour of the

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A. Gordon et al. / Dynamics o f ferroelectric interphase boundaries 211

interphase boundaries in high magnetic fields is of special interest.

This paper is a continuation of ref. [35] in which we give a more general description of the temperature and magnetic field kinetics of fer- roelectric and antiferroelectric phase transitions. However, we consider here the strongly under- damped motion of the interphase boundary. We study the case of a large kinetic energy density coefficient under the influence of temperature and magnetic fields including the case of high- temperature superconducting perovskites.

where F is the Landau-Khalatnikov transport coefficient which sets the time scale of the relaxation process and is assumed to depend noncritically on temperature; the term including the functional derivative OF/OP is one tending to restore the value of P to its thermal equilibrium value; F is given by eqs. (1) and (2). However, this equation gives the overdamped motion of the interphase boundary. To consider a more general case, we take into account the kinetic energy density. Then, the Euler-Lagrange equa- tion is given as

2. Thermo-induced dynamics of ferroelectric interphase boundaries

We start from the Ginzburg-Landau function- al of the total free energy for uniaxial ferroelec- trics (this expression may also be applied for quasi-one-dimensional hydrogen-bonded fer- roelectrics; it may be used when we are inter- ested in dielectric properties along one of the crystal axes):

F[P(x, t)l = f k \ Ox ] (1)

where P is the polarization, f ( P ) is the free- energy density for a uniform system undergoing a first-order phase transition

f ( P ) = ½aP 2 - ¼ b e 4 + i c e 6 (2)

where coefficients b and c are positive, D is the positive coefficient of the inhomogeneity term, coefficient a is temperature-dependent: a = a ' ( T - T o ) , T O is the stability limit of the paraelectric phase, coefficient a ' does not de- pend on temperature.

Until now, the dynamics of ferroelectric inter- phase boundaries has been considered with the help of the time-dependent Ginzburg-Landau equation for the evolution of spontaneous polari- zation [36]. In this case we have the following equation:

OP OF Ot - F OP (3)

02p OZp 2 D Ox 2 - p ~ t 2 - a P + b P 3 - c P 5 = 0 . (4)

The Lagrangian density L is L = K - F, where K is the kinetic energy. We take K as

x { O P ] 2

We present the kinetic energy as the energy of the oscillations of ions. It is assumed that the polarization in a ferroelectric is due to the displacement of a definite ion [37]: P = n e z and the mass density p = r n / n e 2, where z is the displacement of the ion, e is the effective charge, m is the effective mass and n is the number of ions in a unit volume. For the sake of simplicity we neglect the energy of elastic oscillations and elastic energy in eq. (5). The role of elastic effects on the interphase boundary dynamics was studied in ref. [6].

Taking into account the damping term with ( 1 / F ) ( O P / O t ) we obtain the following equation of motion:

OZP O2P 1 OP 2 D Ox z - p Ot 2 I" Ot a P + b P 3 - c P 5 = 0 .

(6)

For the mass density p = 0 we have the usual time-dependent Ginzburg-Landau equation de- scribing the overdamped motion of the inter- phase boundary [38,39]. Substituting s = x - v t

into eq. (6) we have

212 A. Gordon et al. / Dynamics of ferroelectric interphase boundaries

d s 2 +

=0 .

dP v - - ~ - F ( a P - b P 3 + c P 5)

(7)

The partial solution of eq. (7) is known [38,2]. The ferroelectric interphase boundary is given by

p = P o l V l

where

2 b( P0 = ~-c 1

+ e x p ( - ~ ) (8)

+ 4ac - b2 i . (9)

Here we have new expressions for the interphase boundary width A and its velocity v:

3

{ D },,2

x a ' ( T - To)[1 - 3~/8 + (1 - 3~/4)"21(1 + pq~/2) '

pq2,~ 1t2 o=qD1/2/ 1 + - - T - ) ,

where

(10)

(11)

To)] 1/2F i 6 - (2/3)(1 + %/1- 36/4)/] q = 2/"[a'(Tc - L ~ 1 - ~ 3 ~ % / 1 ~ J

(12)

with

T - T o 6 - r c - T~" (13)

Here T¢ is the phase transition temperature. The dependence of the interphase boundary

width A on the dimensionless temperature 6 is shown in fig. 1. This width A is given in units of 3 r ~ [ D / a ( T ¢ - T 0 ) ] 1/2 for different values of the mass density p. Starting from the definite value of p, the temperature dependence of the inter- phase boundary width becomes the curve with maximum, while the width is an increasing function of temperature in the Ginzburg-Landau theory [38,18] and in ref. [35] for small p. In fig. 2 the velocity v of the interphase boundary is

A 2 y 1.8-

1,6-

1 .4 -

1 ~ . - ~ 1

0.8 0.6" 0.4- 02- 0

\

o o~ o:, o'.6 o18 1 112 ,., 8

Fig. 1. The temperature dependence of the interphase boundary width A. The width is given in units of 3[D/a'(Tc - To)] l~z as a function of the dimensionless tem- perature ~ = ( T - To) / (T ~ - To).

V

0.04

0.02

-0.02

-0.04 ) 0 0'.2 0', 0'.8 0'8 ', 1'2 , ,

6 Fig. 2. The temperature dependence of the interphase boundary velocity o. The velocity is presented in units of 2F[Da ' (T c - To)] 1~2 as a function of the dimensionless tem- perature 6 for large values of the mass density p.

presented as a function of 6 for large mass density p. We see the sharp change of the temperature dependence of the velocity v: for large values of the mass density p the tempera- ture curve tends to a saturation and the velocity decreases compared to the case of p = 0 [38,18] and for small p in ref. [35]. The velocity o is given in units of 2 F [ a ' ( T c - To)] 1/2. The expres- sion (10) is different from the one obtained for

A. Gordon et al. / Dynamics of ferroelectric interphase boundaries 213

the overdamped case [38,18], in which pqZ/2 ~ 1, where we obtain the known expression for the interphase boundary width A [38,18]:

D }1/2

A=-43 a ' (Tc - T0)[1 - 3a/8 + V1 - 3a/4]

(14)

In the same limiting case (pq2 /2 ~ 1) the velocity of the interphase boundary v (eq. (11)) is given by

2 r D - - -~- + v --~-

- F ( a P - b P 3 + c P 5 + g P H z + h P H 4 ) = 0 . (17)

In this case (at T = T c, i.e. for 6 = 1) the inter- phase boundary has the width

D (18) A = y 1 + p x 2 / 2

where

v = qD 1/2 . (15)

Equation (15) is the known expression for the interphase boundary velocity [38,18].

The solution (8) was also derived for the nonlinear lattices with dissipation in refs. [40] and [41] (stable, self-sustained lattice kinks in- trinsic to the dissipated lattice system). The inertia effects for different types of nonlinear lattices were taken into account in ref. [40]. This solution is a kink-type solitary wave which pre- sents the interphase ' boundary separating paraelectric and ferroelectric phases. It describes the propagation of the interphase boundary leading to the phase transition.

3. Magnetic-field-induced dynamics of ferroelectric interphase boundaries

We shall consider the dynamic aspects of the phase transition as a growth in the presence of an applied magnetic field H. To study the mag- netic-field induced dynamics of the interphase boundary, we add the terms containing the magnetic field influence to the free energy den- sity. Then dynamic equations (6) and (7) are given as

02P OZP 1 OP

2D ~ x 2 - p Ot 2 l" Ot

- aP + b P 3 - cP 5 - g P H 2 - h P H 4 = 0 (16)

Fb (1 + B - ~/1 - B)

x - (6c)1/2 ~(5 - B ) / 4 + X/1 - B (19)

(6c) 1/2 1 , (20)

Y - b ~ ( 5 - B ) / 4 + ~ / 1 - B

16c B = 7 (gH2 ÷ h H 4 ) ' (21)

and moves with velocity

D (22) o = x 1 + p x 2 / 2 "

According to ref. [37], D ~ d 2, where d is the lattice parameter. Since the lattice parameter d is not changed under the influence of the applied magnetic field, coefficient D is independent of magnetic field strength. The transport coefficient F does not include the dependence of the phase transition temperature [36]. For this reason we assume that the magnetic-field dependence of F is negligible.

Figures 3 and 4 show the magnetic field dependences of the width A and velocity v for BaTiO 3. Here g = a'a and h = a'/3, where a ' = 6.7× 10-SK -1 [42], a =6 .27× 10 -4KT -2 [33], / 3 = 6 . 2 8 x 1 0 - 4 K T -2 [33]. We use b = 9 . 7 x 108 V mS/C 3 [42] and c = 3.9 x 101° V m9/C 5 [42]; the magnetic field strength is given in tesla (T). For the strong damping (pX2/2 "~ 1) we have

A = y D 1/2 (23)

and and

214 A. Gordon et al. / Dynamics of ferroelectric interphase boundaries

A 1

0.8-

0.6.

0.4

0.21 0

o s , 0 ,'~ :~o :;s 30 14 (7")

Fig. 3. The magnetic field dependence of the interphase boundary width A for BaTiO 3 in the case of large values of the mass density p. The width is given in units of (6Dc)'12/b.

V

0.0014

0.001

0.0006

0.000:~

o ~ ,o ~ ~o 2'~ 30 H ( I )

Fig. 4. The magnetic field dependence of the interphase boundary velocity v for BaTiO 3 in the case of large values of the mass density p. The velocity is presented in units of Fb(D/Gc) ll2.

v = x D 1/2 . (24)

In figs. 3 and 4 the magnetic field dependence of the interphase boundary width A and the velocity o are shown for large values of the mass density p. The width A is given in units of ( 6 c D ) ~ / 2 / b .

As is known [35] in the overdamped case the width is widened when the applied magnetic field increases. Large values of the inertia coefficient

lead to the decrease of the interphase boundary width (fig. 3). For small values of p the velocity of the interphase boundary is an increasing function of the applied magnetic field [35] while fig. 4 presents the interphase boundary velocity as a function of the external magnetic field for large values of p showing the tendency to a saturation for large magnetic fields. Therefore, the change of magnetic field dependence takes place for the width and for the velocity at large values of the inertia coefficient. The velocity u is given in units of 2 F b ( D / 6 c ) l /2.

4. Discussion

We have used here the data for perovskites because their thermal interphase boundary dy- namics has been well studied [1,3-5,43-47]. As is known [10], some high-temperature supercon- ducting materials are ferroelectric perovskites. Perhaps the BCS electron pairing is induced by soft phonons in the ferroelectric perovskites due to the considerable enhancement of the elec- tron-phonon interaction in inherently unstable lattices. The intimate relation between high-tem- perature superconductivity and ferroelectricity has extensively been studied in recent years [7- 12]. It has recently been shown [48] that fer- roelectric phase transitions in some high-tem- perature superconductive perovskites are first- order ones. The similarity between normal and superconducting perovskites shows that this de- scription of the temperature and magnetic field kinetics of ferroelectric first-order phase transi- tions can also be effective for the ferroelectric superconductive perovskites.

The analogous consideration of phase transi- tion kinetics can be carried out for the Kittel model [42] of antiferroelectricity. As was shown in ref. [5], the interphase boundary of the Kittel antiferroelectric may also be expressed as a kink solution (8). Consequently, the results presented above may be applied for antiferroelectrics.

Since the movement of the interphase bound- ary is related to the growth of ferroelectric and antiferroelectric crystals, the magnetic field ef- fect can be used to govern the growth processes.

A. Gordon et al, / Dynamics of ferroelectric interphase boundaries 215

5. Summary

This paper has examined the thermo- and magnetic-field-induced dynamics of the ferro- electric interphase boundaries. The growth process accompanying the symmetry-breaking first-order ferroelectric phase transition is associ- ated with the propagation of the interphase boundaries separating the paraelectric and the ferroelectric phases. In this case we have studied the propagation of the interphase boundary under the influence of the temperature and external magnetic field for large inertia coeffi- cients. It has been shown that the temperature and magnetic field dependences of the width and the velocity of ferroelectric interphase boundaries are essentially different from those calculated within the framework of the Ginz- burg-Landau theory and of the small mass density case. The obtained results are applied for ferroelectric and antiferroelectric perovskites in- cluding ferroelectric high-temperature supercon- ductive perovskites.

Acknowledgements

The authors are grateful to Dr. S. Dorfman for useful discussions. This research was sup- ported by the German-Israel Foundation for Scientific Research and Development, Grant N, G-112-279.7/88.

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